Numerous hotels with varying prices in south mumbai area.
If Trident is beyond budget the following hotels can be considered :
The Abode, Colaba
Suba Palace, Colaba
Hotel Suncity Colaba
Fort Residency, Fort
Hotel Regency Colaba
Chetau Windsor, Churchgate
All of them are reasonably priced and conveniently lacerated from tourism perspective.
Places like Paharganj in Delhi are not easy to find in Mumbai. At least not at those cheap prices for sure. Everything in Mumbai is more expensive, especially hotel rooms. Some areas that do offer cheap accommodation are not in very safe and clean areas so would not recommend staying in such places. Better to stay in slightly better hotels in areas like Fort and Colaba which offer different types of budget and good hotels at reasonable prices and good locations.
Hope this helps.
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Also the recommendations given by Car Link regarding the budget hotels are really nice and helpful!!